# Theoretical Ordinary Differential Equation with the lim Yg = 0

I was given a theoretical problem in my Linear Algebra and Differentials class and I am stumped on how to approach this problem.

Consider the ODE y''+ay'+by=0 where a and b are constants such that $$a^2$$ < 4b. What conditions on a and b guarentees that the general solution of the ODE satisfies $$\displaystyle{\lim_{x \to \infty} Yg=0}$$

So I do understand that Yg is the general solution, which contains the homogeneous solution + the particular solution.

So far we have worked with simple polynomials in our class, so what we do is we solve for the particular solution first and use $$y=C1e^(r1*x) + C2e^(r2*x)$$ where t1 and t2 are the roots to the equation $$r^2+ar+b=0$$ (from above equation).

The way we solve the particular solution is by "Guessing" the right hand side of this equation, but since it is = 0 I believe that the Yg is just the Y homogeneous.

$$Yg = Yh+Yp => Yg = Yh+0 => Yg=Yh$$

So in this case I believe my Yg would be of the form $$Yg = e^(r1*x) + e^(r2*x)$$ which implies that my lim equation is $$\displaystyle{\lim_{x \to \infty} e^(r1*x) + e^(r2*x)=0}$$

I am not sure if this is right at all, but this is how I would currently approach it. I just have no idea how to satisfy the constraint given of $$a^2<4b$$ and that the limit of Yg is = 0.

Any guidance would be greatly appreciated.